#8531654

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,39)(36,40)(37,38);(3,7)(5,8)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,16,18,15)(13,19,21,20)(17,23,24,22)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8438850);